In general it's not injective, nor surjective. It's just a map.
For example, if $M = S^n$, the sphere. $\pi_0 Diff(M)$ is the group of homotopy $(n+1)$-spheres, but $Out(\pi_1 M)$ is trivial. So you have surjectivity but not injectivity provided exotic spheres exist in that dimension. There's lots of much more complicated examples of this type, for example $M = S^1 \times D^n$, this goes back to work of Farrell, Hatcher, Quillen, Igusa.
I think you mean to formulate the question a little differently since manifolds can have orientiation-reversing diffeomorphisms and you rarely "capture" that by $Out(\pi_1)$. For example, $\pi_0 Diff(S^2) \simeq \mathbb Z_2$ yet $Out(\pi_1 S^2)$ is trivial.
But similarly, $\pi_0 Diff(S^2 \times S^1) \simeq \mathbb Z_2^2$ and $Out(\pi_1 S^2 \times S^1)$ is $\mathbb Z_2$. But even in the orientation-preserving case, there's no isomorphism.
edit: okay if you're specifically interested in $K(\pi,1)$ spaces, there's the computation of Hatcher -- $\pi_0 Diff( (S^1)^n)$ is an extension of $GL_n \mathbb Z$ by:
$$ \mathbb Z_2^\infty\oplus\binom n2\mathbb Z_2\oplus\sum_{i=0}^n\binom n i\Gamma_{i+1} $$
That's an infinite direct-sum of $\mathbb Z_2$'s together with a direct sum of many groups of exotic spheres. See the Wikipedia page.